Solving a class of multi-scale elliptic PDEs by means of Fourier-based mixed physics informed neural networks

Deep neural networks have received significant attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this work, we probe into solving a class of elliptic PDEs with multiple scales by means of Fourier-based mixed physics-informed neural networks (called FMPINN), and its solver is configured as a multi-scale DNN model. Unlike the classical PINN method, a dual (flux) variable about the rough coefficient of PDEs is introduced to avoid the ill-condition of neural tangent kernel matrix that resulted from the oscillating coefficient of multi-scale PDEs. Therefore, apart from the physical conservation laws, the discrepancy between the auxiliary variables and the gradients of multi-scale coefficients is incorporated into the cost function, then leveraging the optimization method to yield the satisfactory solution of PDEs by minimizing the defined loss. Additionally, a novel trigonometric activation function is introduced for FMPINN, which is suited for representing the derivatives of complex target functions. Handling the input data by Fourier feature mapping will effectively improve the capacity of deep neural networks to solve high-frequency problems. Finally, by introducing several numerical examples of multi-scale problems in various dimensional Euclidean spaces, we validate the efficiency and robustness of the proposed FMPINN algorithm in both low-frequency and high-frequency oscillation cases.